Over the past fifteen years, a new class of refrigerators and heat engines has been developed. These devices utilize intrinsically irreversible thermal conduction or acoustical geometry to provide the proper phasing between pressure and volumetric velocity. This phasing produces useful quantities of heat pumping or cooling or generates mechanical work. These new engines are called thermoacoustic engines. Earlier engines required mechanical means, such as pistons, linkages, displacers, cams, valves and other mechanisms to realize useful cooling or produce mechanical work using more traditional reversible heat engine cycles, e.g., Stirling Cycle or Rankine Cycle. The heat pumping power of these new acoustic refrigerators depends upon the square of the acoustic amplitude--a doubling of acoustic pressure amplitude corresponds to four times the useful heat-pumping power. It is therefore important to be able to produce very high amplitude sound waves for use in such refrigeration devices.
Over the last decade, a new fluid pump that employs high-amplitude acoustic standing waves has also been developed, U.S. Pat. No. 5,319, 938 and U.S. Pat. No. 5,515,684. These sound waves actuate reed valves that rectify oscillatory pressure to produce the mean flow and to permit continuous, unidirectional pumping of gases by the high-amplitude sound waves. A very high-power sound source is necessary to make this "sonic compressor" effective, and a highly efficient conversion from electrical to acoustical energy is required to make it economical.
Nearly all of the electrically driven thermoacoustic refrigerators patented and/or produced to date have used a moving-coil, electrodynamic loudspeaker to generate the required high-amplitude sound waves. These moving coil loudspeakers had several attractive features associated with their relatively low moving (dynamic) mass. The low mass meant that a fairly flexible suspension could be used to provide a high resonance frequency, usually in the range of several hundred Hertz. This lower moving mass also permitted the operating frequency of the thermoacoustic refrigerator to be adjusted over a modest range of frequencies to allow the system to be tuned over a small bandwidth without substantially degrading efficiency. Unfortunately, the moving-coil loudspeaker efficiency and power handling capacity is limited by the mass of conductor (typically copper, aluminum, or copper-cladded aluminum) in the coil.
The development of high flux density rare-earth magnetic materials (e.g., NdFeB) and the recent invention of a high-efficiency linear alternator which uses such magnets for mechanical-to-electrical power conversion, has made it practical to consider a moving-magnet electrodynamic system as a possible high-amplitude, high-efficiency electrodynamic sound source. The Yarr/Corey design, U.S. Pat. No. 5,389,844, uses several coils wound around a multi-pole laminated magnetic stator to increase the available mass of conductor by several orders-of-magnitude over the moving-coil loudspeaker without directly affecting the length of the magnetic gap or adding to the moving mass which, in the new moving-magnet configuration, is controlled by the mass of the moving rare-earth magnets.
An additional advantage to making the coil part of the magnetic stator is that the electrical leads which bring current to the coil do not have to flex to accommodate motion ofthe moving-magnet part of the motor. The flexure of the input current leads in a moving-coil electrodynamic motor is a fairly common cause of motor failure which the moving-magnet design avoids, thereby increasing its reliability over the moving-coil design.
To appreciate the utility of increasing the operating (mechanical resonance) frequency of such a moving-magnet driver by increasing its resonance frequency by the methods taught herein, one need only to examine the expression for the time-averaged acoustic power, .PI..sub.ac, supplied by a moving piston of area, A.sub.piston : ##EQU1##
The variables sub-scripted with a "1" are the peak values of quantities which are assumed to have a sinusoidal time variation with a frequency f=.omega./2.pi.. Observing that convention, the piston stroke (peak-to-peak displacement amplitude) is given by 2d.sub.1. Further calculation is simplified by the introduction of the piston's peak volumetric velocity, U.sub.1 [m.sup.3 /sec].
The above expression, Eq. 1, includes the simplifying assumptions that the piston speed, V.sub.1, and the net force on the face of the piston, F.sub.1, are in-phase. Although there are many circumstances of practical interest for which this assumption is not valid, the assumption is true for a piston that is driving an acoustic load which is oscillating at one of its acoustic resonance frequencies. A resonant acoustic load is commonly found in thermoacoustic refrigerators and sonic compressors.
The acoustic pressure amplitude, p.sub.1, can be related to the volumetric velocity of the piston, U.sub.1, by the introduction of an acoustic impedance, Z.sub.ac [Newton-sec/m.sup.5 ], which is given by the ratio of the pressure to the volumetric velocity at the piston location. ##EQU2##
The acoustic impedance is a function only of the acoustic load presented to the piston and not a function of the drive mechanism.
Substitution of Eq. 2 into Eq. 1 demonstrates that the acoustic power delivered by the piston to the resonator characterized by Z.sub.ac, is dependent upon the square of the product of the piston displacement, d.sub.1, and the radian frequency, .omega. [rad/sec], of the piston's oscillation. ##EQU3##
It is clear that the full exploitation of moving-magnet electrodynamic motors for the generation of high-amplitude sound fields, with high electroacoustic efficiency, requires that the motors operate at the highest possible frequency, .omega., while maintain the ability to utilize their maximum allowable stroke, 2d.sub.1. Since the resonance frequency of such a motor, having a moving mass, m.sub.0 [kg], is determined by the total suspension stiffniess, k [Newton/m], ##EQU4##
it is desirable to increase the total suspension stiffness k without undue restriction on the motor stroke and with an effectively infinite fatigue lifetime for the spring.
The most common approach for providing auxiliary stiffness to increase the resonance frequency of a moving-magnet motor, commonly employed by the Stirling-Cycle engine community, is the use of a "gas spring." The advantage to the gas spring is that there are no material limitations, such as fatigue fracture, when gas pressure provides the additional restoring force (stiffness). One disadvantage to the gas spring is that there is irreversible thermal dissipation due to the adiabatic heating and cooling of the gas which accompanies the gas spring's compressions and expansions. This dissipation mechanism increases the motor's effective mechanical resistance, R.sub.m, in addition to increasing its net suspension stiffness, k. Another disadvantage is that the gas spring stiffness is also dependent upon mean pressure, p.sub.m, in the "back volume" which creates the gas spring, so the driver mechanical resonance will change if the mean operating pressure is changed.
An even more serious limitation to the gas-spring approach is that pressure behind the piston and in front of it are 90.degree. out-of-phase if the driver is located at the pressure anti-node of the standing wave, or nearly 180.degree. out-of-phase for the displaced driver location (located closer to the main portion of the resonator), so the piston seal (either a clearance seal or bellows) must accommodate a larger front-to-back pressure differential, .DELTA.p [Pa], if a large spring constant is required, as is the case for moving-magnet motors. This excess front-to-back pressure differential will complicate the bellows design due to the required trade-off between the infinite-lifetime bellows excursion (stroke) and the pressure differential. It will also increase dissipation in a clearance seal, since "blow-by" losses increase proportionate to the square of the front-to-back pressure differential, (.DELTA.p).sup.2.
An advantage of gas springs is that their design is fairly straightforward. In addition to the front of the piston do ing work on the engine, the rear of the piston is used to adiabatically compress the gas in the back-volume, V.sub.0 [m.sup.3 ], behind the piston. This gas spring can provide a restoring force wi th an equivalent stiffniess, k.sub.gas [Newtons/m], which is related to the area of the piston of diameter D, A.sub.piston =.pi.D.sup.2 /4, the mean gas pressure, P.sub.m, the ratio of specific heat at constant pressure to specific heat at constant volume, .gamma.=c.sub.p /c.sub.v, and the volume of the gas trapped behind the piston, V.sub.o : ##EQU5##
The disadvantage of this design process is the requirement that V.sub.o be very small if a large gas stiffness is required. "Dead volume," required to accommodate the gas-filled volumes which contain essential components such as the coils, plunger, bellows, pole pieces, etc., limit how small the volume, V.sub.o, can be made in practice.
Another prior-art approach for providing auxiliary stiffness to increase the resonance frequency of a moving-magnet motor is to use the uniform width rectangular cantilever leaf spring design. The leaf spring is composed of a beam which is has uniform length, L, width, w, and thickness, t. One end of the beam is clamped and its position is held fixed, while the other end is guided, i.e. clamped (forcing the slope of the spring to be zero at both clamping locations), but the position ofthe movable end of the beam can displace in a direction perpendicular to the clamping plane. Its transverse displacement from its equilibrium (F=0) position is designated, y.sub.A [m].
The spring constant, k.sub.rect, of the rectangular clamped-guided cantilever is the ratio of the force, F, to the displacement of the movable end, y.sub.A [m], generated by that force applied to the movable end. For the rectangular cantilever of length, L [m], width, w [m], and thickness, t [m], ##EQU6##
where E [Pa] is the Young's modulus of the spring material. The maximum material stress, .sigma..sub.max [Pa], occurs at both the upper and lower surfaces of the spring, which are at the maximum distance from the neutral axis (.+-.t/2). ##EQU7##
The above result for .sigma..sub.max can be used to express the spring constant in terms of the ratio of the maximum allowable stress, based on maximum stress, .sigma..sub.fat [Pa], dictated by the material fatigue considerations, to the Young's modulus, E. ##EQU8##
The form of Eq. 8 above is useful because the ratio of the maximum allowable stress in a material to its Young's modulus characterizes the fatigue properties of the material, its heat treatment history and its surface condition. It also shows that for a given material, the spring must be longer to accommodate a larger displacement if the maximum allowable stress in the material, .sigma..sub.fat, is not to be exceeded. The thickness of such a spring is also determined by imposition of the maximum allowable stress constraint. ##EQU9##